A geometric realization of the chromatic symmetric function of a unit interval graph

Shareshian-Wachs, Brosnan-Chow, and Guay-Pacquet [Adv. Math. $\mathbf{295}$ (2016), $\mathbf{329}$ (2018), arXiv:1601.05498] realized the chromatic (quasi-)symmetric function of a unit interval graph in terms of Hessenberg varieties. Here we exhibit another realization of these chromatic (quasi-)symmetric functions in terms of the Betti cohomology of the variety ${\mathcal{X}}_{\mathrm{\Psi }}$ defined in [arXiv:2301.00862]. This yields a new inductive combinatorial expression of these chromatic symmetric functions. Based on this, we propose a geometric refinement of the Stanley-Stembridge conjecture, whose validity would imply the Shareshian-Wachs conjecture.